Proof

We have to give a 1-page proof showing that the list we have constructed of all closed, oriented 2-manifolds (sphere, torus, 2-hole torus, 3-hole torus, etc....) is a complete list without repetitions of all closed, oriented 2-manifolds.

So I have to show that the list is complete and I have to show that there are no repetitions.

We have to give a 1-page proof showing that the list we have constructed of all closed, oriented 2-manifolds (sphere, torus, 2-hole torus, 3-hole torus, etc....) is a complete list without repetitions of all closed, oriented 2-manifolds.

So I have to show that the list is complete and I have to show that there are no repetitions.

According to the classification theorem of closed surfaces (2-manifold), any closed surface is homeomorphic to some member of one of these three families.

For 1 and 2, a surface that is the connected sum of n tori is said to be of genus n, while a sphere is of genus 0 (Intuitively speaking, a genus number corresponds a number of handles in a closed orientable surface).

This genus number is also involved in Euler characteristic, which is topologically invariant. So we can say that we classify closed orientable 2-manifolds up to homeomorphism by using a genus.